Two concentric coils, each of radius equal to \( 2\pi \) cm, are placed at right angles to each other. Currents of 3 A and 4 A respectively are flowing through the two coils. The magnetic induction at the center of the coils will be
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 50\,\mu\mathrm{T} \)
Each coil produces at the centre a field \( B = \mu_{0}I/(2r) \), directed along the axis of that coil. The axes are at right angles, so the two fields are perpendicular and add as vectors.
\( \displaystyle r = 2\pi\,\mathrm{cm} = 2\pi\times10^{-2}\,\mathrm{m} \)
\( \displaystyle B_{1} = \frac{\mu_{0}I_{1}}{2r} = \frac{4\pi\times10^{-7} \times 3}{2 \times 2\pi\times10^{-2}} = 3\times10^{-5}\,\mathrm{T} = 30\,\mu\mathrm{T} \)
\( \displaystyle B_{2} = \frac{4\pi\times10^{-7} \times 4}{2 \times 2\pi\times10^{-2}} = 4\times10^{-5}\,\mathrm{T} = 40\,\mu\mathrm{T} \)
\( \displaystyle B = \sqrt{B_{1}^{2} + B_{2}^{2}} = \sqrt{30^{2} + 40^{2}} = 50\,\mu\mathrm{T} \)